In geological exploration it is desirable to obtain information regarding the various formations and structures that exist beneath the Earth's surface. Such information may include determine geological strata, density, porosity, composition, etc. This information is then used to model the subsurface basin using the obtained data to predict the location of hydrocarbon reserves and aid in the extraction of hydrocarbon.
Unstructured grids have many appealing characteristics for modeling physical processes in complex geologic structures such as subsurface basins. Such grids may also be used in other industries, for example in the airspace industry or the auto industry. The basin or domain of interest may be modeled or represented as a set of layers of different thickness stacked together. The geological layers may be fractured along vertical or slanted surfaces and degenerate creating so-called pinch-outs. Pinch-outs are defined as parts of geological layers with near zero thickness. This complexity should be taken into account by the grid to produce a good model of the geological layers. An unstructured grid provides a better model than a structured grid. An unstructured grid may comprise a set of polyhedral elements or cells defined by their vertices and have a completely arbitrary topology. For example, a vertex of the grid can belong to a number of cells and each cell can have any number of edges or faces.
Many physical subsurface processes may be described by mathematical equations of convection-diffusion type. Examples of such processes can be fluid flow in porous media, temperature distribution, and/or pressure distribution. An important process for oil exploration is the temperature distribution or thermal modeling. Thermal modeling involves the heat moving from the magma below the crust and through the sedimentary layers and source rock. Source rocks are rocks that are involved in the formation of oil and other hydrocarbon materials. The oil and/or other hydrocarbon materials would be expelled from the source rocks and migrate elsewhere. The quality of hydrocarbon is determined by the temperature and pressure conditions inflicted on the source rocks and their surrounding area. The quality is also affected by the temperature and pressure conditions of the migration path between the source rocks and its current location. Thus, the pressure and temperature conditions of the basis throughout its history is important.
To more accurately model the processes, it is important to model not only the primary variables, such as pressure or temperature, but also model their fluxes, or the rates of flow of energy, fluids, etc. over any given surface There is a variety of known approaches for modeling these processes, such as finite difference, finite volume, or finite element methods. In these approaches, where a physical process is considered, the domain is covered by a grid. Then, the domain is approximated on a grid by introducing a set of unknowns called the degrees of freedom at specified locations of the grid cells and deriving algebraic equations for each location that connect the degree of freedom in that location with other degrees of freedom. The way of deriving such equations, as well as the locations of degrees of freedom, is different for different approaches mentioned above, but all these methods have a common feature, namely, that they only involve primary variables, such as temperature or pressure.
To compute fluxes, an interested person would first compute the desired primary variable using one of the above-described approaches, and then use numerical differentiation to compute the flux of the primary variable. All existing methods of numerical differentiation being accurate on regular grids, e.g. rectangular or parallelepiped grids, are inaccurate and very computationally expensive on unstructured grids, especially if the domain where the physical process is considered is highly heterogeneous. Moreover, the approaches for solving convection-diffusion problems using finite difference methods require Cartesian grids, and thus are not applicable in many subsurface applications, which have to employ unstructured grids. The finite element methods, being able to model complex geometries do not have local conservation property and can not be applied in many subsurface processes. Conversely, finite volume approaches are locally conservative and can be applied on a subset of unstructured grids which are locally orthogonal. However, when the unstructured grid does not posses local orthogonality property, finite volume method provides inaccurate solution. Thus, from all three classes of the approaches mentioned above, none is applicable for description of subsurface convection-diffusion processes in a basis modeled with an unstructured grid.
There is another mathematical approach to simultaneously approximate primary unknowns and their fluxes, called mixed finite element method, which is described in F. Brezzi and M. Fortin, “Mixed and hybrid finite element methods”, Springer Verlag, Berlin 1991. Such method is proven to be locally mass conservative, accurate in the presence of heterogeneous medium, and provide accurate approximations to both, primary unknowns and fluxes. Until recently, the mixed finite element methods could not be directly applied to the domains covered by unstructured polyhedral grids, which are very common for the subsurface applications. A new version of mixed finite element method for diffusion-type equations on arbitrary polyhedral grids is proposed in Yu. Kuznetsov and S. Repin, “New mixed finite element method on polygonal and polyhedral meshes”, Russian Journal of Numerical Analysis and Mathematical Modeling, v. 18, pp. 261-278, 2003.